Question

If the coefficients of 2nd, 3rd and the 4th terms in the expansion of ${(1+x)}^n$ are in A.P, then value of $n$ is

• A. $5$
• B. $7$
• C. $11$
• D. $14$

Answer 1

The correct option is B $7$

For the terms to be in A.P, it must follow
$(n-2r{)}^2=n+2$
In the above case $r=2$
Substituting in the equation, we get
$(n-4{)}^2=n+2$
$n^2-8n+16=n+2$
$n^2-9n+14=0$
$(n-2)(n-7)=0$
$n=2$ and $n=7$
However for $n=2$ we will have only $3$ terms.
Hence the required answer is $7.$